On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.

Computing locating-total domination number in some rotationally symmetric graphs

Let [Formula: see text] be a connected graph. A locating-total dominating set in a graph G is a total dominating set S of a G, for every pair of vertices [Formula: see text], such that [Formula: see text]. The minimum cardinality of a locating-total dominating set is called locating-total domination number and represented as [Formula: see text]. In this paper, locating-total domination number is determined for some cycle-related graphs. Furthermore, some well-known graphs of convex polytopes from the literature are also considered for the locating-total domination number.

Heptavalent Symmetric Graphs with Certain Conditions

A graph [Formula: see text] is said to be symmetric if its automorphism group [Formula: see text] acts transitively on the arc set of [Formula: see text]. We show that if [Formula: see text] is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group [Formula: see text] of automorphisms, then either [Formula: see text] is normal in [Formula: see text], or [Formula: see text] contains a non-abelian simple normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups. If [Formula: see text] is arc-transitive, then [Formula: see text] is always normal in [Formula: see text], and if [Formula: see text] is regular on the vertices of [Formula: see text], then the number of possible exceptional pairs [Formula: see text] is reduced to 5.

On 2-metric resolvability in rotationally-symmetric graphs

The 2-metric resolvability is an extension of metric resolvability in graphs having several applications in intelligent systems for example network optimization, robot navigation and sensor networking. Rotationally symmetric graphs are important in intelligent networks due to uniform rate of data transformation to all nodes. In this article, 2-metric dimension of rotationally symmetric plane graphs Rn, Sn and Tn is computed and found to be independent of the number of vertices.

Symmetric graphs of valency seven and their basic normal quotient graphs

We characterize seven valent symmetric graphs of order 2 p q n 2p{q}^{n} with p < q p\lt q odd primes, extending a few previous results. Moreover, a consequence partially generalizes the result of Conder, Li and Potočnik [On the orders of arc-transitive graphs, J. Algebra 421 (2015), 167–186].